I am currently on a L1-A visa working for Company A. Parallel, I would like to start/work for my own business. Is this possible?

I hear this question a lot - a real testament to the entrepreneurial spirit of the people coming to the U.S.

Thing is, the answers get complicated and the best bet is to speak with a lawyer to figure out your options. I can provide a few general principals.

We don't necessarily make it easy for people to start up new companies here unless they are willing to "bet the farm" and give up the day job. Not many have that kind of money.

You can passively invest money, as you would in stocks, but to do any productive work for the benefit of the new entity requires a visa permitting that work. Just about anything you would do to promote the company, provide managerial direction or guidance, etc. would be considered work - just not taking a salary doesn't change this.

There are the E-1/E-2 treaty trader/treaty investor visas, which require a decent investment and wouldn't let you keep working at your current employer (plus, your home country has to have a treaty in place with the U.S. for these - since you are from Europe, probably not a problem). But overall, this doesn't help.

In fact, since you can't have two separate visas at a time, and your L-1A limits you to a specific type of work for the L-1A employer. Close to anything you would do here would violate the terms of the L-1A

You ask about incorporating the company abroad and doing work for it here; not sure this helps you. There are a few things that an employee of an overseas company can come here briefly to do for the employer abroad without needing a visa authorizing employment (basically, using a visitor visa): attend meetings or conferences, negotiate contracts, even perform after-sales service pursuant to a contract the overseas company has with a U.S. company. But, you wouldn't be on a visitor visa, and aren't talking a bout a short visit.

Here's one thought: while you can't have two different visa types at the same type, there is a type of professional worker visa where you can have two of this visa at the same time ("concurrently"). You may be able to do your current job on this visa and get a separate one for your own company (both tricky propositions in their own right, but not impossible.

Another: where is the current employer on petitioning for a green card for you? You may be eligible for a faster path to a green card than most, and before even getting it would be eligible for "open market" employment authorization that lets you work in any capacity, without worrying about violating the terms of the L.

Speak to a lawyer about the possibilities - often there is a way to find solutions.

Good luck!!